eighth of a sphere with extremely high face count
Well this all sounds highly suspicious and unnecessary, but if you must.

Start with a Bezier Circle object.
After adding it enter edit mode and change all handle types to either Free or Aligned.

Afterwards erase the necessary vertex so only one quarter remains.

Once done add a Screw modifier to it and change the axis to $Y$, and adjust the angle to 90º.



You can now non destructively adjust the Subdivison levels on both directions, either from the Screw Modifier steps parameter or from the curve resolution..

You can maintain a lower level for both for viewport display purposes so it doesn't slow down to much, and higher levels for final rendering.

Edit:

You can achieve a similar result as above replacing the bezier curve circle by a single vertex mesh, then using two screw modifiers on different axis, overcoming the limits of curve resolution.



This has the added benefit of yielding a mesh base object (which is more flexible and better supported) and it is easier to control since all parameters can be controlled from the modifier stack